EXAMPLE: SECTION 7.1: INTEGRATION BY PARTS (THE ANTI-PRODUCT RULE)

110.109 CALCULUS II (PHYS SCI & ENG) PROFESSOR RICHARD BROWN

∫ Question 1. Calculate sin−1 x dx.

Note. This is an example of a integral does not have the form of a product of functions. However, by thinking of one of the functions as a constant 1, the Integration by Parts procedure allows us to instead integrate the derivative of the inverse sine function. This will be easier. One other point is that this function sin−1 x is NOT “1 over the sine function”, or 1 sin−1 x ≠ = sec x. sin x This is a common mistake due to ambiguous notation. Rather, it is the function-inverse of the sine function, as in y = sin−1 x is the inverse function of sin y = x. To solve this, we will first compute the derivative of sin−1 x. Then we will use this fact. Proof. To compute the derivative of the function y = cos−1 x, we will instead use the inverse of this function sin y = x and use implicit differentiation. Here then thinking of y as an implicit function of x, we have d d (sin y) = (x) dx dx dy (cos y) = 1 dx dy 1 = . dx cos y This is not quite complete, as we have only expressed the derivative of y = sin−1 x in terms of the function y. We can fix this through an understanding of how triangles relate the trigonometric functions of their angles to their side lengths and such. Form a right triangle with one angle y and hypotenuse 1. Then the side adjacent to the angle y has length cos y and the side opposite to the angle y has length sin y. Use the picture at the end as a guide, and notice that√ the opposite side sin y = x. The Pythagorean’s Theorem gives us that x2 + cos2 y = 1, or cos y = 1 − x2. Thus dy d ( ) 1 = sin−1 x = √ . dx dx 1 − x2 Why was this helpful? Now use Integration by Parts to rewrite the integral. Here, let f(x) = sin−1 x, and g(x) = 1 in the formula ∫ ∫ f(x)g′(x) dx = f(x)g(x) − f ′(x)g(x) dx.

1 Then f ′(x) = √ , and g(x) = x, and we get 1 − x2 ∫ ∫ x sin−1 x dx = x sin−1 x − √ dx. 1 − x2 1 2 110.109 CALCULUS II (PHYS SCI & ENG) PROFESSOR RICHARD BROWN

To finish the calculation, notice that the integral on the right hand side of the last equation can be − 2 − − 1 solved with a straightforward substitution: Let u = 1 x , so that du = 2x dx, or 2 du = x dx. Then ∫ ∫ x sin−1 x dx = x sin−1 x − √ dx − 2 ∫ 1 x −1 = x sin−1 x − √ du 2 u √ √ = x sin−1 x + u + C = x sin−1 x + 1 − x2 + C. 

[ √ ] d Is this correct? Let’s check by showing that x sin−1 x + 1 − x2 + C = sin−1 x. Here dx

[ √ ] ( ) d x 1 x sin−1 x + 1 − x2 + C = sin−1 x + √ + √ (−2x) + 0 dx 1 − x2 2 1 − x2 x x = sin−1 x + √ − √ 1 − x2 1 − x2 = sin−1 x.

1 cos y

y sin y = x